Mercer's Theorem - Trace

Trace

The following is immediate:

Theorem. Suppose K is a continuous symmetric non-negative definite kernel; TK has a sequence of nonnegative eigenvalues {λi}i. Then

This shows that the operator TK is a trace class operator and

Read more about this topic:  Mercer's Theorem

Famous quotes containing the word trace:

    “Muhammad is the Messenger of God,
    and those who are with him are hard
    against the unbelievers, merciful
    one to another. Thou seest them
    bowing, prostrating, seeking bounty
    from God and good pleasure. Their
    mark is on their faces, the trace of
    prostration....
    God has promised
    those of them who believe and do deeds
    of righteousness forgiveness and
    a mighty wage.”
    —Qur’An. Victory 48:35, ed. Arthur J. Arberry (1955)

    “Beauty is a precious trace that eternity causes to appear to us and that it takes away from us. A manifestation of eternity, and a sign of death as well.”
    —Eugène Ionesco (b. 1912)

    ““In your company a man could die,” I said, “a man could die and you wouldn’t even notice, there’s no trace of friendship, a man could die in your company.””
    —Max Frisch (1911–1991)